Process Rate Estimator
A modeling side-hustle for the ETH group sustainable agroecosystems
1 Introduction
Denitrification is the natural process by which nitrate (NO3-) in the soil are converted by bacteria into nitrous oxide (N2O) or pure nitrigen (N2). The latter is called total denitrification — the full process described in Equation 1 takes place.
\[ \ce{NO3^- ->[\text{Nitrate}][\text{reductase}] NO2^- ->[\text{Nitrite}][\text{reductase}] NO ->[\text{Nitrite oxide}][\text{reductase}] N2O^- ->[\text{Nitrous oxide}][\text{reductase}] N2} \tag{1}\]
Denitrification occurs in conditions where oxygen is limited, such as waterlogged soils. It is part of the nitrogen cycle, where nitrogen is circulated between the atmosphere, organisms and the earth.
2 Formal model description
2.1 Model parameters
| Symbol | Code | Name | Value | Unit |
|---|---|---|---|---|
| \(BD\) | BD |
Bulk density (mass of the many particles of the material divided by the bulk volume) | \(1.686\) | g cm-3 |
| \(\theta_w\) | theta_w |
Soil volumetric water content | ||
| \(\theta_a\) | theta_a |
Air-filled porosity | \(1-\frac{\theta_w}{\theta_t}\) | |
| \(\theta_t\) | theta_t |
Total soil porosity | \(1-\frac{BD}{2.65}\) | |
| \(\text T\) | temperature |
Soil temperature | \(298\) | K |
| \(D_{\text{s}}\) | D_s |
Gas diffusion coefficient | Equation 3 | m2s-1 |
| \(D_{\text{fw}}\) | D_fw |
Diffusivity of N2O in water | Equation 5 | |
| \(D_{\text{fa}}\) | D_fa |
Diffusivity of N2O in air | Equation 6 | |
| \(D_{\text{fa,NTP}}\) | Free air diffusion coefficient under standard conditions | Equation 6 | ||
| \(n\) | n |
Empirical parameter (1) | 1.81 | |
| \(H\) | H |
Dimensionless Henry’s solubility constant | Equation 4 | |
| \(\rho\) | rho |
Gas density of N2O | \(1.26 \times 10^6\) | mg m-3 |
The diffusion fluxes between soil increments are described by Frick’s law (Equation 2).
\[F_{\text{calc}} = \frac{dC}{dZ} D_{\text s} \rho \tag{2}\]
Here, \(D_s\) is the gas diffusion coefficient, \(\rho\) is the gas density of N2O, and \(\frac{dC}{dZ}\) is the N2O concentration gradient from lower to upper depth. The fluxes are calculated based on N2O concentration gradients between 105-135 cm, 75-105 cm, 45-75 cm, 15-45 cm, and 0-15 cm depth layers, and ambient air above the soil surface.
\(\theta_w\) is the soil volumetric water content, \(\theta_a\) the air-filled porosity, and \(\theta_T\) is the total soil porosity.
The gas diffusion coefficient \(D_{\text s}\) was calculated according Equation 3 as established by Millington and Quirk in 1961 (2).
\[D_{\text s} = \left( \frac{\theta_w^{\frac{10}{3}} + D_{\text fw}}{H} + \theta_a^{\frac{10}{3}} \times D_{\text fa} \right) \times \theta_T^{-2} \tag{3}\]
Here, \(H\) represents a dimensionless form of Henry’s solubility constant (\(H'\)) for N2O in water at a given temperature. The constant \(H\) for N2O is calculated as follows:
\[H = \frac{8.5470 \times 10^5 \times \exp \frac{-2284}{\text T}}{\text R \times \text T} \tag{4}\]
Here, \(\text R\) is the gas constant, and \(\text T\) is the temperature (\(\text T = 298 \; \text K\)).
\(D_{\text{fw}}\) was calculated according to Equation 5 as documented by Versteeg and Van Swaaij (1988) (3).
\[D_{\text{fw}} = 5.07 \times 10^{-6} \times \exp \frac{-2371}{\text T} \tag{5}\]
\[D_{\text{fa}} = D_{\text{fa, NTP}} \times \left( \frac{\text T}{273.15} \right)^n \times \left( \frac{101'325}{\text P} \right) \tag{6}\]
2.2 Smoothing curves
The N2O concentration, site preference as well as \(\delta\)18O are estimated as a function of time for every depth and every column, separately. To achieve this function approximation, Kernel Regression as implemented in npreg is used (4). Besides choosing a Kernel, the model only requires a single hyperparameter, i.e. the bandwidth (bws), which facilitates the hyperparameter tuning.
